Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques.

Dweik, Samer; Santambrogio, Filippo

In this paper we consider the mass transportation problem in a bounded domain Ω where a positive mass f in the interior is sent to the boundary ∂Ω. This problems appears, for instance in some shape optimization issues. We prove summability estimates on the associated transport density σ, which is the transport density from a diffuse measure to a measure on the boundary f− = P#f (P being the projection on the boundary), hence singular. Via a symmetrization trick, as soon as Ω is convex or satisfies a uniform exterior ball condition, we prove Lp estimates (if f ∈ Lp, then σ ∈ Lp). Finally, by a counter-example we prove that if f ∈ L∞ (Ω) and f− has bounded density w.r.t. the surface measure on ∂Ω, the transport density σ between f and f− is not necessarily in L∞ (Ω), which means that the fact that f− = P#f is crucial.

SUMMABILITY theory; DIRICHLET problem; BOUNDARY value problems; STRUCTURAL optimization; MATHEMATICAL singularities

ESAIM: Control, Optimisation & Calculus of Variations, 2018, Vol 24, Issue 3, p1167

1292-8119

Academic Journal

10.1051/cocv/2017018